From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5609 Path: news.gmane.org!not-for-mail From: Ronnie Brown Newsgroups: gmane.science.mathematics.categories Subject: Re: Composing modifications Date: Thu, 04 Mar 2010 07:24:56 +0000 Message-ID: <4B8F6048.7000903@btinternet.com> References: Reply-To: Ronnie Brown NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1267729464 754 80.91.229.12 (4 Mar 2010 19:04:24 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 4 Mar 2010 19:04:24 +0000 (UTC) To: David Leduc Original-X-From: categories@mta.ca Thu Mar 04 20:04:19 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1NnGLL-0007me-5m for gsmc-categories@m.gmane.org; Thu, 04 Mar 2010 20:04:19 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NnFmP-0002RD-C3 for categories-list@mta.ca; Thu, 04 Mar 2010 14:28:13 -0400 User-Agent: Thunderbird 2.0.0.23 (Windows/20090812) In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5609 Archived-At: Dear All, Has anyone formulated a cubical (square?) version of bicategory or higher? The model is surely the singular cubical complex KX of a topological space. This has a great deal of structure of multiple compositions and tensor products which we have fairly easily written down and exploited in papers with Philip Higgins. By working with filtered spaces and taking certain homotopy classes we also get (non trivially!) associated strict structures. There is also a notion of fibrant (=Kan) cubical set, and of fibration, which seems not available globularly (?). So one formulation of a weak cubical omega-category is to say it comes with a cubical fibration to a strict cubical omega-category. (Such exists for RX_*, the cubical singular complex of a filtered space. (with P.J. HIGGINS), ``Colimit theorems for relative homotopy groups'', {\em J. Pure Appl. Algebra} 22 (1981) 11-41. ) This is a definition with one example and no theorems (as yet)! By contrast, there is a singular globular complex GX of a space, see for example `A new higher homotopy groupoid: the fundamental globular $\omega$-groupoid of a filtered space', Homotopy, Homology and Applications, 10 (2008), No. 1, pp.327-343. but I think nobody has written down an axiomatisation. Multiple compositions are difficult (for me, at any rate) in the globular (and simplicial!) situation, so I tend to prefer the simple minded approach. I have spent many happy hours subdividing squares by horizontal and vertical lines into lots and lots of little squares! Ronnie David Leduc wrote: > Dear Nick and Tom, > > Thank you very much for your replies. It is very helpful. > > I had in mind to form a tricategory of bicategories, therefore I guess > I was talking of what Tom calls strong transformations. They are also > called weak??? I am a bit confused by the lax, weak, pseudo, strict > and so on terminology in higher category theory. Nick, could you > confirm that you were talking of strong transformations in your mail? > Now I am not sure anymore what are natural transformations in category > theory. They are strict transformations, right? > > Unfortunately, I do not have a copy of the paper "Coherence for > tricategories" by Gordon, Power and Street. I guess such reference > would help me a lot with such questions. > > I have another question. For strong and strict (and maybe lax?) > transformations, we have the interchange law relating vertical and > horizontal composition. What is the equivalent of interchange law for > compositions of modifications? > > Thank you, > > David [For admin and other information see: http://www.mta.ca/~cat-dist/ ]